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Full text of "Treatise On Analysis Vol-Ii"

3    POSITIVE MEASURES: THE ABSOLUTE VALUE OF A MEASURE       107

By virtue of (13.3.2.1) it is enough to show that |ju(/)| ^ ji(l/l) fr al*
/e JTC(X). Given/, there exists  e C such that || = I and |X/)| = M/) =
; since /* is real, we have MC/) = A<#(f/)) >' and since /j ^ 0 and
 |/|, it follows that /i(#(f/)) ^ Ml/I), which establishes the
assertion.

If jw is any raz/ measure on X, it follows from (13.3.3) that fi g |ju|, and
therefore :

(13.3.6)   Every real measure on X is the difference of two positive measures
(for a more precise result, see (13.15)).

If /* is any (complex) measure on X, it follows from (13.3.3) and (13.3.2)
that we have

(13.3.7)

Also if ^, v are any two measures on X we have

(13.3.8)                                    |/ + v|  |/| + |v|

by virtue of (13.3.2).

Finally, it follows immediately from the definitions that, if n : X - X' is a
homeomorphism and ^ is any measure on X, then

(13.3.9)                                      brGOl -

PROBLEMS

1. Let X be a locally compact space, E a vector subspace of ^R(X) and P a convex cone
in ^R(X) (i.e., a subset of this space such that the relations/ep and #ep imply
/+ g e P and afe P for all scalars a > 0). Suppose that for each function h e Jf R(X)
there exists /e E such that / h e P,

Let if be a real linear form on E such that the relation/e E n P implies u(f) > 0.
Let h e JfR(X) and let Pi be the set of all /e E such that h ~/E P, and P' the set of
all/e E such that/ h e P. Show that these two sets Pi, P are nonempty, and that if
of is the least upper bound of the (/) with/e Pi and a" the greatest lower bound of
the u(f) with /e Pi', then a' and a" are finite and a" <J a". Deduce that there exists a
linear form i on the subspace E! * E + Rh of ^R(X) which extends w, such that the
relation /i e EI n P implies w^C/i) j> 0. Show that a' g u^(h) <J a" for any such exten-
sion i of u> and that the extension is unique if and only if of = a*.*. is a positive measure on X, then